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I'm trying to solve questions regarding bayesian network, and now I was wondering if it is possible to know the probability of an unknown variable in the tree. For instance, I have this tree,

A         B
\         /
 \       /
  \     /
   \   /
     C

Very simple, I have A = 0.1 and A'=0.9, B=0.1 and B'=0.9 Is it possible by any way to know the probability of C given A and B ?

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No, $P(C|A,B)$ can't be determined solely from the marginal distributions $P(A)$ and $P(B)$. For example, say $C$ perfectly determines $A$ and $B$, via one of two mechanisms:

  1. If $C=1$, then $A=1$ and $B=1$ (and the marginal distribution of $C$ is the same as $A$ and $B$: $P(C=1) = 0.1$). In this case, $P(C=1|A=1, B=1)=1$.

  2. If $C=1$, then $A=0$ and $B=0$ (and the marginal distribution of $C$ is now: $P(C=1) = 0.9$). In this case, $P(C=1|A=1, B=1)=0$.

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